Pre-Algebra Examples

$\frac{\frac{x}{9} - \frac{1}{x}}{1 - \frac{3}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $9 x$ .

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Step 1.1

Multiply $\frac{\frac{x}{9} - \frac{1}{x}}{1 - \frac{3}{x}}$ by $\frac{9 x}{9 x}$ .

$\frac{9 x}{9 x} \cdot \frac{\frac{x}{9} - \frac{1}{x}}{1 - \frac{3}{x}}$

Step 1.2

Combine.

$\frac{9 x (\frac{x}{9} - \frac{1}{x})}{9 x (1 - \frac{3}{x})}$

Step 2

Apply the distributive property.

$\frac{9 x \frac{x}{9} + 9 x (- \frac{1}{x})}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $9$ .

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Step 3.1.1

Factor $9$ out of $9 x$ .

$\frac{9 (x) \frac{x}{9} + 9 x (- \frac{1}{x})}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.1.2

Cancel the common factor.

$\frac{9 x \frac{x}{9} + 9 x (- \frac{1}{x})}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.1.3

Rewrite the expression.

$\frac{x \cdot x + 9 x (- \frac{1}{x})}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.2

Raise $x$ to the power of $1$ .

$\frac{x^{1} x + 9 x (- \frac{1}{x})}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.3

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{1} + 9 x (- \frac{1}{x})}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} + 9 x (- \frac{1}{x})}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.5

Add $1$ and $1$ .

$\frac{x^{2} + 9 x (- \frac{1}{x})}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.6

Cancel the common factor of $x$ .

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Step 3.6.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{x^{2} + 9 x \frac{- 1}{x}}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.6.2

Factor $x$ out of $9 x$ .

$\frac{x^{2} + x \cdot 9 \frac{- 1}{x}}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.6.3

Cancel the common factor.

$\frac{x^{2} + x \cdot 9 \frac{- 1}{x}}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.6.4

Rewrite the expression.

$\frac{x^{2} + 9 \cdot - 1}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.7

Multiply $9$ by $- 1$ .

$\frac{x^{2} - 9}{9 x \cdot 1 + 9 x (- \frac{3}{x})}$

Step 3.8

Cancel the common factor of $x$ .

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Step 3.8.1

Move the leading negative in $- \frac{3}{x}$ into the numerator.

$\frac{x^{2} - 9}{9 x \cdot 1 + 9 x \frac{- 3}{x}}$

Step 3.8.2

Factor $x$ out of $9 x$ .

$\frac{x^{2} - 9}{9 x \cdot 1 + x \cdot 9 \frac{- 3}{x}}$

Step 3.8.3

Cancel the common factor.

$\frac{x^{2} - 9}{9 x \cdot 1 + x \cdot 9 \frac{- 3}{x}}$

Step 3.8.4

Rewrite the expression.

$\frac{x^{2} - 9}{9 x \cdot 1 + 9 \cdot - 3}$

Step 3.9

Multiply $9$ by $- 3$ .

$\frac{x^{2} - 9}{9 x \cdot 1 - 27}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

$\frac{x^{2} - 3^{2}}{9 x \cdot 1 - 27}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{(x + 3) (x - 3)}{9 x \cdot 1 - 27}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $9$ out of $9 x \cdot 1 - 27$ .

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Step 5.1.1

Factor $9$ out of $9 x \cdot 1$ .

$\frac{(x + 3) (x - 3)}{9 (x \cdot 1) - 27}$

Step 5.1.2

Factor $9$ out of $- 27$ .

$\frac{(x + 3) (x - 3)}{9 (x \cdot 1) + 9 \cdot - 3}$

Step 5.1.3

Factor $9$ out of $9 (x \cdot 1) + 9 \cdot - 3$ .

$\frac{(x + 3) (x - 3)}{9 (x \cdot 1 - 3)}$

Step 5.2

Multiply $x$ by $1$ .

$\frac{(x + 3) (x - 3)}{9 (x - 3)}$

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

$\frac{(x + 3) (x - 3)}{9 (x - 3)}$

Step 6.2

Rewrite the expression.

$\frac{x + 3}{9}$